1 Abstract

Foraging behavior shapes animal health through the balance of the resources expended to obtain nutrients and the energetic rewards of captured resources. The health of individuals will influence reproductive success, fitness and ultimately population viability. Despite the key role of individual health in evolutionary ecology, it can be challenging to connect foraging behavior and energetics in free living marine organisms. Combining data on foraging strategies, feeding behavior and physiology, we estimate body mass for humpback whales as they forage near the Antarctic Peninsula. Humpback whales require extreme energetic storage to facilitate the yearly migration between austral foraging areas to equatorial breeding grounds. Based on tagged individuals, we model the foraging effort, duration and energetics of animals throughout the austral season. We then connect these measurements to studies of whale feeding physiology and biomechanics. While predictive modeling of individual behavior will be naturally complex, we utilize hierarchical Bayesian models to propagate uncertainty throughout the model. By testing a variety of model assumptions surrounding energetics and foraging rate, we can understand the sensitivity of our predictions to parameter bounds. Our model forms the basis for field tests of animal body condition and further exploration of individual behavior and animal health. Future work combining the environmental impacts of changing ocean conditions on foraging success will inform conservation strategies in a changing marine ecosystem.

2 Foraging Model

The proportion of time an animal is in a feeding behavioral state.

Process Model

\[Y_{i,t+1} \sim Multivariate Normal(d_{i,t},σ)\]

\[d_{i,t}= Y_{i,t} + γ_{s_{i,g,t}}*T_{i,g,t}*( Y_{i,g,t}- Y_{i,g,t-1} )\]

\[ \begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix} \] \[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\] The behavior at time t of individual i on track g is a discrete draw. \[S_{i,g,t} \sim Cat(\phi_{traveling},\phi_{foraging})\]

Dive information is a mixture model based on behavior (S)

\(\text{Average dive depth}(\psi)\) \[ \psi \sim Normal(dive_{\mu_S},dive_{\tau_S})\]

Dive profiles per indidivuals

sink(“Bayesian/Diving.jags”) cat(" model{

pi <- 3.141592653589

#for each if 6 argos class observation error

for(x in 1:6){

##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]

#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
logit(phi[i,g,t,1]) <- alpha[state[i,g,t-1]]  + beta * dive[i,g,t]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement in location
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)

}

#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha[state[i,g,steps[i,g]-1]] - beta * dive[i,g,steps[i,g]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])

}
}
}
}

###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016

##Behavioral States

gamma[1] ~ dbeta(3,2)       ## gamma for state 1
dev ~ dbeta(1,1)            ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev

#Intercepts
alpha[1] ~ dbeta(1,1)
alpha[2] ~ dbeta(1,1)

#Effect of dive depth on foraging state
beta ~ dnorm(0,0.0001)

#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd

#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539

#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156

}"
,fill=TRUE)

sink()

##     user   system  elapsed 
##    1.236    0.160 3766.565

Foraging Model

Ben Weinstein

2017-10-31 12:22:32

1 Abstract

2 Foraging Model

2.1 Chains

2.2 Temporal autocorrelation in foraging

2.2.1 Time foraging as a function of time

3 Temporal Variation in Dive Behavior

3.1 Diel

3.2 Month

4 Comparison with 2d movement model

4.1 Chains